A novel single-loop meta-model importance sampling with adaptive Kriging for time-dependent failure probability function

Abstract

To learn the effect of interested distribution parameter, also the design variable of random input vector, on time-dependent failure probability, and to decouple time-dependent reliability-based design optimization (T-RBDO), estimating time-dependent failure probability function (T-FPF), a relation of time-dependent failure probability varying with the distribution parameter in interested design region, is necessary. However, estimating T-FPF is time-consuming and a challenge at present. Thus, this paper proposes a novel single-loop meta-model importance sampling with adaptive Kriging model (SL-Meta-IS-AK) to estimate T-FPF efficiently. In SL-Meta-IS-AK, for estimating the T-FPF by single-loop simulation, an optimal importance sampling probability density function (IS-PDF), which can envelope the interested distribution parameter region and be free of the distribution parameter, is constructed by an integral operation. After the Kriging model is adaptively constructed for time-dependent performance function to approach optimal IS-PDF for T-FPF by quasi-optimal one, a simple sampling strategy is designed to extract the samples of quasi-optimal IS-PDF, and a time-dependent misclassification probability function is derived to update the Kriging model adaptively until it can accurately recognize the states of all extracted samples, on which the T-FPF at the whole interested distribution parameter region can be estimated as a byproduct. Due to the single-loop simulation aided by the IS-PDF covering the interested distribution parameter region but free of the distribution parameter, the efficiency of estimating T-FPF is improved by the proposed SL-Meta-IS-AK, which is verified by presented numerical and aviation engineering examples including a wing structure and a turbine shaft structure.

Appendices

Appendix 1: Kriging surrogate model

Select training set ${\varvec{T}} = \{ ({\varvec{x}}_{1} ,g({\varvec{x}}_{1} )),({\varvec{x}}_{2} ,g({\varvec{x}}_{2} )), \cdots ,({\varvec{x}}_{{N_{t} }} ,g({\varvec{x}}_{{N_{t} }} ))\}^{T}$ ($N_{t}$ is the size of the training set) by a design of experiment (DOE) for performance function $Y = g({\varvec{x}})$. Then, the Kriging model $g_{K} ({\varvec{x}})$ of $g({\varvec{x}})$ can be established by Kriging toolbox as:

$$g_{K} ({\varvec{x}}) = {\varvec{f}}^{T} ({\varvec{x}}){\varvec{\beta}} + Z({\varvec{x}})$$

(42)

where ${\varvec{f}}^{T} ({\varvec{x}}){\varvec{\beta}}$ is the regression model. ${\varvec{f}}({\varvec{x}}) = [f_{1} ({\varvec{x}}),f_{2} ({\varvec{x}}), \cdots ,f_{n} ({\varvec{x}})]^{T}$ is the basis vector of the regression function, and $n$ represents the number of basis functions. ${\varvec{\beta}} = (\beta_{1} ,\beta_{2} , \cdots ,\beta_{n} )^{T}$ is the coefficient vector of the regression function. $Z({\varvec{x}})$ is a Gaussian process with a mean of zero and a standard deviation of $\sigma$. Its covariance matrix is:

$$Cov[Z({\varvec{x}}_{i} ),Z({\varvec{x}}_{j} )] = \sigma^{2} R({\varvec{x}}_{i} ,{\varvec{x}}_{j} )$$

(43)

where $R({\varvec{x}}_{i} ,{\varvec{x}}_{j} )$ is the correlation function of any two sample points, and it has many expressions. In this paper, the following Gaussian form is adopted,

$$R({\varvec{x}}_{i} ,{\varvec{x}}_{j} ,{\varvec{\xi}}) = \exp \left( { - \sum\limits_{k = 1}^{n} {{\varvec{\xi}}_{k} \left| {x_{i}^{(k)} - x_{j}^{(k)} } \right|^{2} } } \right)$$

(44)

where $x_{i}^{(k)}$ represents the $k$-dimensional component of the sample ${\varvec{x}}_{i}$. ${\varvec{\xi}} = (\xi_{1} ,\xi_{2} , \cdots ,\xi_{n} )^{T}$ is an unknown correlation parameter vector, and it can be obtained by maximum likelihood estimation as follows,

$$\hat{\varvec{\xi }} = \arg \mathop {\min }\limits_{{\varvec{\xi}}} \left[ {\left| {\varvec{R}} \right|^{{1/N_{t} }} \hat{\sigma }^{2} } \right]$$

(45)

The regression coefficient vector ${\varvec{\beta}}$ and the variance $\sigma^{2}$ of the Gaussian process can be obtained from the training points shown in Eqs. (46) and (47), respectively,

$$\hat{\varvec{\beta} } = \left( {\varvec{F}^{T} \varvec{R}^{ - 1} \varvec{F}} \right)^{ - 1} \varvec{F}^{T} \varvec{R}^{ - 1}\varvec{ y}$$

(46)

$$\hat{\sigma }^{2} = \frac{1}{{N_{t} }}\left( {\varvec{y} - \varvec{F}\hat{\varvec{\beta} }} \right)^{T} \varvec{R}^{ - 1} \left( {\varvec{y} - \varvec{F}\hat{\varvec{\beta} }} \right)$$

(47)

where ${\varvec{F}}$ is an $N_{t} \times n$ regression matrix shown as follows:

$$\varvec{F = }\left[ \begin{gathered} f_{1} ({\varvec{x}}_{1} ) f_{2} ({\varvec{x}}_{1} ) \cdots f_{n} ({\varvec{x}}_{1} ) \\ f_{1} ({\varvec{x}}_{2} ) f_{2} ({\varvec{x}}_{2} ) \cdots f_{n} ({\varvec{x}}_{2} ) \\ \vdots \vdots \ddots \vdots \\ f_{1} ({\varvec{x}}_{{N_{t} }} ) f_{2} ({\varvec{x}}_{{N_{t} }} ) \cdots f_{n} ({\varvec{x}}_{{N_{t} }} ) \\ \end{gathered} \right]$$

(48)

${\varvec{R}}$ is an $N_{t} \times N_{t}$ correlation matrix in Eq. (49).

$$\varvec{R = }\left[ \begin{gathered} R({\varvec{x}}_{1} ,{\varvec{x}}_{1} ) R({\varvec{x}}_{1} ,{\varvec{x}}_{2} ) \cdots R({\varvec{x}}_{1} ,{\varvec{x}}_{{N_{t} }} ) \\ R({\varvec{x}}_{2} ,{\varvec{x}}_{1} ) R({\varvec{x}}_{2} ,{\varvec{x}}_{2} ) \cdots R({\varvec{x}}_{2} ,{\varvec{x}}_{{N_{t} }} ) \\ \vdots \vdots \ddots \vdots \\ R({\varvec{x}}_{{N_{t} }} ,{\varvec{x}}_{1} ) R({\varvec{x}}_{{N_{t} }} ,{\varvec{x}}_{2} ) \cdots R({\varvec{x}}_{{N_{t} }} ,{\varvec{x}}_{{N_{t} }} ) \\ \end{gathered} \right]$$

(49)

According to the principle of Kriging model, the predicted value at any untried point ${\varvec{x}}$ follows the Gaussian distribution with a mean of $\hat{\mu }_{{g_{K} }} ({\varvec{x}})$ and a variance of $\hat{\sigma }_{{g_{K} }}^{2} ({\varvec{x}})$; namely, $\hat{g}({\varvec{x}}) \sim N\left( {\hat{\mu }_{{g_{K} }} ({\varvec{x}}),\hat{\sigma }_{{g_{K} }}^{2} ({\varvec{x}})} \right)$. $\hat{\mu }_{{g_{K} }} ({\varvec{x}})$ and $\hat{\sigma }_{{g_{K} }}^{2} ({\varvec{x}})$ are shown in Eqs. (50) and (51), respectively,

$$\hat{\mu }_{{g_{K} }} (\varvec{x}) = \varvec{f}^{T} (\varvec{x})\hat{\varvec{\beta} } + \varvec{r}^{T} (\varvec{x})\varvec{R}^{ - 1} (\varvec{y} - \varvec{F}\hat{\varvec{\beta} })$$

(50)

$$\hat{\sigma }_{{g_{K} }}^{2} (\varvec{x}) = \hat{\sigma }^{2} [1 + \varvec{u}^{T} (\varvec{x})(\varvec{F}^{T} \varvec{R}^{ - 1} \varvec{F})^{ - 1} \varvec{u}(\varvec{x}) - \varvec{r}^{T} (\varvec{x})R^{ - 1} \varvec{r}(\varvec{x})]$$

(51)

where ${\varvec{r}}({\varvec{x}})$ is the correlation coefficient vector between training sample and predicted points. ${\varvec{r}}({\varvec{x}})$ and ${\varvec{u}}({\varvec{x}})$ are shown in Eqs. (52) and (53), respectively.

$${\varvec{r}}({\varvec{x}}) = [R({\varvec{x}},{\varvec{x}}_{1} ),R({\varvec{x}},{\varvec{x}}_{2} ), \cdots ,R({\varvec{x}},{\varvec{x}}_{{N_{t} }} )]^{T}$$

(52)

$$\varvec{u}(\varvec{x}) = \varvec{F}^{T} \varvec{R}^{ - 1} \varvec{r}(\varvec{x}) - \varvec{f}(\varvec{x})$$

(53)

Appendix 2: the proof of the importance samples

Suppose that the cumulative distribution function of the random vector ${\varvec{X}}$ is $F_{{\varvec{X}}} ({\varvec{a}}) = P\{ X_{1} \le a_{1} ,X_{2} \le a_{2} , \cdots ,X_{n} \le a_{n} \} = P\{ {\varvec{X}} < {\varvec{a}}\}$. Then,

$$F_{{\varvec{X}}} ({\varvec{a}}) = P\{ {\varvec{X}} < {\varvec{a}}\} = \int_{{ - {\mathbf{\infty }}}}^{{\varvec{a}}} { \cdots \int {\hat{h}_{{\varvec{X}}} (\varvec{x|\theta })d{\varvec{x}}} } = \frac{{\int_{{ - {\mathbf{\infty }}}}^{{\varvec{a}}} { \cdots \int {\pi_{{F_{t} }} (\varvec{x|\theta })f_{{\varvec{X}}} (\varvec{x|\theta })d{\varvec{x}}} } }}{{P_{ft\varepsilon } ({\varvec{\theta}})}}$$

(54)

Under the condition of the acceptance domain of $\Omega = \{ p - c\pi_{{F_{t} }} (\varvec{x|\theta }) \le 0\}$, where $p\sim U[0,1]$, $\varvec{x\sim }f_{{\varvec{X}}} (\varvec{x|\theta })$ and $c\pi_{{F_{t} }} (\varvec{x|\theta }) \le 1$, the conditional CDF $F_{{{\varvec{X}}|\Omega }} ({\varvec{a}}) = P\{ {\varvec{X}} < {\varvec{a}}|p < c\pi_{{F_{t} }} (\varvec{x|\theta })\}$ of vector ${\varvec{X}}$ on $\Omega$ can be estimated as follows:

$$\begin{gathered} F_{{{\varvec{X}}|\Omega }} ({\varvec{a}}) = P\{ {\varvec{X}} < {\varvec{a}}|p < c\pi_{{F_{t} }} (\varvec{x|\theta })\} \\ = \frac{{P\{ {\varvec{X}} < {\varvec{a}},p < c\pi_{{F_{t} }} (\varvec{x|\theta })\} }}{{P\{ p < c\pi_{{F_{t} }} (\varvec{x|\theta })\} }} \\ = \frac{{\int_{{ - {\mathbf{\infty }}}}^{{\varvec{a}}} { \cdots \int {\int_{0}^{{c\pi_{{F_{t} }} ({\varvec{a}})}} {f_{P} (p)f_{{\varvec{X}}} (\varvec{x|\theta })dpd{\varvec{x}}} } } }}{{\int_{{ - {\mathbf{\infty }}}}^{{ + {\mathbf{\infty }}}} { \cdots \int {\int_{0}^{{c\pi_{{F_{t} }} ({\varvec{a}})}} {f_{P} (p)f_{{\varvec{X}}} (\varvec{x|\theta })dpd{\varvec{x}}} } } }} = \frac{{\int_{{ - {\mathbf{\infty }}}}^{{\varvec{a}}} { \cdots \int {\pi_{{F_{t} }} (\varvec{x|\theta })f_{{\varvec{X}}} (\varvec{x|\theta })d{\varvec{x}}} } }}{{\int_{{ - {\mathbf{\infty }}}}^{{ + {\mathbf{\infty }}}} { \cdots \int {\pi_{{F_{t} }} (\varvec{x|\theta })f_{{\varvec{X}}} (\varvec{x|\theta })d{\varvec{x}}} } }} \\ = \frac{{\int_{{ - {\mathbf{\infty }}}}^{{\varvec{a}}} { \cdots \int {\pi_{{F_{t} }} (\varvec{x|\theta })f_{{\varvec{X}}} (\varvec{x|\theta })d{\varvec{x}}} } }}{{P_{ft\varepsilon } ({\varvec{\theta}})}} \\ \end{gathered}$$

(55)

where $f_{P} (p) = \left\{ \begin{gathered} 1, \, 0 \le p \le 1 \hfill \\ 0, {\text{else}} \hfill \\ \end{gathered} \right.$ is PDF of the standard uniform random variable $p$.

According to Eqs. (54) and (55), it is obvious that $F_{{\varvec{X}}}^{{\tilde{h}}} ({\varvec{a}}) = F_{{{\varvec{X}}|\Omega }}^{f} ({\varvec{a}})$. Therefore, the samples generated from the acceptance domain $\Omega$ are those generated from $\hat{h}_{{\varvec{X}}} (\varvec{x|\theta })$.